The boolean Pythagorean Triples problem has a positive answer

You need to know: Notation for the set of positive integers.

Background: A Pythagorean triple is a set of three positive integers a, b, and c, such that .

The Theorem: On 3rd May 2016, Marijn Heule, Oliver Kullmann, and Victor Marek submitted to arxiv a paper in which they proved the following result. The set can be partitioned into two parts, such that no part contains a Pythagorean triple, while this is impossible for .

Short context: The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory. It asks whether in every partition of the set of positive integers into two parts, one can always find a Pythagorean triple in one of the parts. There was a progress in the variants of the problem (see here and here), but the original problem remained open. The Theorem provides a positive answer. In fact, it guarantees that one can find a Pythagorean triple in every partition of the set of first 7825 positive integers, and that 7825 is the smallest integer with this property. The computer-assisted proof is almost 200 terabytes in size.

Links: Free arxiv version of the original paper is here, journal version is here.

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